Minimizing the energy of maps from a surface into a 2-sphere with prescribed degree and boundary values
Let \(S \subset \mathbb{R}^ n\) be an embedded smooth 2-sphere, and let \(\Sigma\) be a compact Riemann surface with nonempty boundary. For a given boundary datum \(\nu \in \text{Lip} (\partial \Sigma, S)\) consider the class \(E = \{u \in W^{1,2} (\Sigma, \mathbb{R}^ n) : u|_{\partial \Sigma} = \nu\), \(u(z) \in S\) for a.e. \(z \in \Sigma\}\). Let \(E_ k = \{u \in E: \text{deg}(u,u_ 0) = k\}\), where \(u_ 0 \in E\) is a fixed map, \(\text{deg}(u,u_ 0)\) is the degree of \(u\) relative \(u_ 0\). There is considered the question: \[ \text{Is } \lambda_ k = \inf \{E(u) = 2^{- 1} \cdot \int_ \Sigma (| u_ x|^ 2 + | u_ y|^ 2 dx dy : u \in E_ k\} \text{ attained?}\tag{1} \] If \(E\) contains a holomorphic (antiholomorphic) map \(u^ +\) (\(u^ -\) respectively), then let \(\text{deg}(u^ +, u_ 0) = k^ +\) \((\text{deg} (u^ -, u_ 0) = k^ -\)). Otherwise set \(k^ + = + \infty\), \(k^ - = - \infty\). The following result is proved. Theorem. \(\lambda_ k\) is attained if and only if \(k^ - \leq k \leq k^ +\). The theorem reduces (1) to a problem in complex function theory.
scientific article; zbMATH DE number 599662
scientific article